further existence results for classical solutions of the equations of a second-grade fluid

Arch. Rational Mech. Anal. 128 (1994) 297-312. �9 Springer-Verlag 1994

Further Existence Results for Classical Solutions of the Equations

of a Second-Grade Fluid

GIOVANNI P. GALDI & ADI~LIA SEQUEIRA

To our dear friend, Professor Christian Simader, on his 50 th birthday

Communicated by K. R. RAJAGOPAL

1. Introduction

Over the last twenty years, there has been a remarkable interest in the study of a class of non-Newtonian fluids, known as fluids of grade n, from both theoretical and experimental points of view, see, e.g., [7, 14, 15, 5, 9, 16, 19] and the references cited therein. For a general incompressible fluid of grade 2, the Cauchy stress T is given by

T = --/~I + #A1 + ~1A2 + c~2A 2, (1.0)

where ~t is the viscosity, ~ , c~2 are material coefficients (normal stress moduli), /~ represents the pressure and A~, A 2 a r e the first two Rivlin-Ericksen tensors [17, 20], defined by

As = Vv + (Vv) r,

d A2 = ~A1 + A1Vv + (vv)r A1.

Here v is the velocity field and ~ denotes the material time derivative. The thermodynamical principles impose some restrictions on e~ and e2 [7]. In particular, the Clausius-Duhem inequality implies that

# > 0 , ~1 + c<2 = 0

and the requirement that the free energy be a minimum in equilibrium implies that

~l=>0.

With these conditions on the stress moduli, the equations of motion for an incompressible homogeneous fluid of grade 2 are given by

~t(v-~Av)-vAv=Vp-curl(v-~Av)XV}v.v=O inf~x(0, T) (1.1)

298 G.P. GALD! & A. SEQUEIRA

(cf. [7]) with the initial and boundary conditions

v ( x , t ) = O on ~3~ x (0, r ) , (1.2)

v(x,O) = Vo(X ) in fL

Here f~ is a bounded domain in R 3 with smooth boundary ~?f~ and ~ x (0, T) (T > 0) is a space-time region. The initial velocity is denoted by Vo(X) and p = !5/p, ~ = el~P, v = #/p, where p is the (constant) density of the fluid. For simplicity, we assume that there are no external body forces acting on the fluid. Under the assumptions

v > 0, ~ > 0,

problem (1.1), (1.2) has been studied by several authors. In particular, CIORANZSCU & EL HAC~NE [5] have given existence and uniqueness results for weak solutions global (in time) in two dimensions, and local (in time) in three dimensions. More recently, GALD~, GROBBELAAR & SAUER [10], by using a different approach, have shown for the general system (even without the restriction cq + ~2 = 0) existence and uniqueness of classical solutions for short time. Furthermore, if the size of the initial data is suitably restricted, their solutions exist for all time. However, for this latter result to hold, it is crucial to require that ~1 ( = eP) is "sufficiently large". It should be remarked that such a condition looks - - in a sense - - artificial, since one expects that global solutions should be favoured by a "large" viscosity and a "small" constant of elasticity. The main objective of the present paper is to remove such a restriction on e. Specifically, we show that if fl is simply-connected and Vo is not too "large", problem (1.1), (1.2) admits one (and only one) classical solution, global in time, for any ~ > 0.1 For the general problem (~1 + 0~2 :~= 0) the steady case has been studied very recently by CoscIa & GALDI [6].

The method we follow here is based on an appropriate splitting of the original problem along with the use of the Schauder fixed-point theorem. The key requirement for this method to work is to obtain sufficiently strong a priori estimates on the higher-order derivatives of the solution. These are obtained by transforming the original problem into an equivalent one as follows. Using the elementary identity

e u r l ( u x v) = v . V u - u ' V v + (V.v)u - (V.u)v (1.3)

with u = e u r l ( v - ~ A v ) (1.4)

and applying the curl operator to the first equation of (1.1) we find that

~u & +-V(ue - c u r l y ) = u . V v - v . V u in f~x(0, T), (1.5)

V - v = 0 in f~ x (0, T), (1.6)

u ( x , 0 ) = cur l (vo - c~Avo) = Uo in f~, (1.7)

vlan = 0 Vte [0, T].

1 The main results of this paper still hold in two space dimensions (plane flow), and, in fact, their proofs are somewhat simpler. We also remark that for a = 0 the problem is well-known since, in this case, equations (1.l) reduce to the Navier-Stokes equations.

A Second-Grade Fluid 299

It is clear that for classical solutions problems (1.1), (1.2) and (1.4)-(1.7) are equivalent. In fact, if u is a solution of (1.4)-(1.7), then from relation (1.3) and the hypothesis on s it follows easily that there exists a function p such that (1.1) holds.

The Schauder theorem is used in the following manner. In a suitable class of functions ~o, we introduce the map

as the composition of the operator ~o ~-+ v, defined by

euri(v - ~Av) = q~ in f~x(0, T),

V . v = O in f~ x (0, T), (1.8)

v I~a = 0

(where the time t is considered as a fixed parameter), with the operator v ~ u defined by

0u v & + - ( u ~ - e u r l v ) = u ' V v - v ' V u in f~x(0 , T),

(1.9) u(.,0) = Uo in ~2.

The existence of a solution to the problem (1.4)-(1.7) (and hence to (1.1), (1.2)) is guaranteed as soon as we are able to show that ~(q~) has a fixed point, which in turn will be achieved by making use of the Schauder theorem.

The paper is organized as follows. In Section 2, we prove existence, uniqueness and a priori bounds for solutions to the linear auxiliary problems (1.8), (1.9). We also derive estimates of the solutions which hold for all times t > 0, provided the size ofuo is suitably restricted. The main results are stated and proved in Section 3. Taking advantage of the a priori estimates and standard compactness arguments, we apply the fixed-point theorem to show local existence in the time interval I-to,to + T ] (to > 0), where T depends solely on an upper bound for the initial data U(to) in the appropriate norm. Successively, we use this result together with the global estimates of Section 2 and a standard "bootstrap" extension argument to prove global existence of classical solutions to the original problem for small data.

2. Unique solvability of some auxiliary problems

To begin with, we introduce some standard notations. By f] we denote a bounded domain of R 3. Throughout the paper we assume that f] is simply- connected, so that any irrotational vector function in ~ is the gradient of some scalar function 2.

2Actually, some of our intermediate results hold under more general assumptions on fl which, in particular, do not require simply-connectedness. However, such an assumption is crucial for Lemma 2.1, e.g., and our main Theorem 3.2 to be valid.

300 G.P. GALDI & A. SEQUEIRA

We follow the usual convention (cf, e.g., El0]) of defining wm'P(~)(Hm(f~), if p = 2) as the s tandard Sobolev space of order m, with the usual no rm II Jim, p( II" tim) and scalar product (',')m,, ((',')m). In H~ = Lz(f~) the norm is denoted by II " I[o, and for simplicity, the scalar product by (','). The space H m- 1/2 (~3f~) equipped with norm II " I[,,- 1/2, ~a is the trace space associated with H " ( ~ ) . By Hm(f~) we denote the Sobolev space of the vector fields v = (vl,vz,v3) such that vi s Hm(f~), i = 1, 2, 3. In the sequel this convent ion will be applied to the other function spaces and norms. Fo r m > 0, we consider the Hilbert spaces V,, = {v e H"( f~) :V-v = 0}, X,, = {v ~ V,,: v .n = 0 on Of~}, where n denotes the unit outer normal vector. They are both closed subspaces of Hm(f~); the norms in V~,X,, are also denoted by II �9 lira. Moreover , for a Banach space Y with no rm ]l']lr and the given time interval I = (to, to + T), to > 0, T > 0, we recall the classical Banach spaces (1 < p < oo )

LP(I; Y) = {v measurable, v:t e I ~ v(t) e Y, S II v(t)II =~dt < oo } I

and wm'P(I; Y), which is the space of functions such that the distributional time derivatives of order up to m are in LP(I; Y). For p = oo, L~(I; Y) is the Banach space of the (measurable) essentially bounded functions defined on I with values in Y. We denote by ]l" Ilk,,,,r the usual norm in Wk'~(I;Hm(f~)), for k > 0 . In particular, for k = 0 we write II'[I 0,m, r = II']]m, r . Finally, the space of functions of class C" on I with values in Y is denoted by cm(I; Y).

The purpose of this section is to study the unique solvability of certain auxiliary problems related to (1.4)-(1.7). We begin by considering the linear problem

eurl(v - ~ A v ) = q~ in fir,

V-v = 0 i n f e r , (2.1)

vlon = 0

for a known function q~, where f~r = f~ x I (the time t is a fixed parameter). We have

Lemma 2.1. Let r ~ Wk'~176 Vm), k, rn > O. Then there exists a unique vector field qt ~ Wk' ~176 (I;Xm+ l ) such that

curlqt = q~, (2.2)

IIq~ I[k,.,+~.T < C H~ol[k,m,r (2.3)

for a constant C = C(m,f~).

Proof. The existence of a unique vector potential ~0 e X~ satisfying curl ~0 = q~ (for r ~ Vo) and the inequality [I ~b Irl =< C I[ q~ rio is proved, for instance, in [12]. The regularity result (2.3) for k = 0 follows easily from [8, Proposi t ion 1.4, page 41]. Finally, differentiating (2.2) k times with respect to t and using uniqueness, one easily sees that O e Wk'~(I;X, ,+I) and that the estimate (2.3) holds for any k__>0. [ ]


Lemma 2.2. For any q~6Wk'~(I;V, , ) , k ,m >=O, there exists a unique v e Wk'~(1; V,,+3) satisfyin9 both (2.1) and the estimate

IlvlJk, m+3, r < Cllq~[lk,,~,r. (2.4)

Proof. By Lemma 2.1 we can write q~ = curl o for some ~ e W k' ~ (I;X,,+ 1)" Let us consider the following Stokes-like problem

v -c~Av + V ~ = ~ in f~r,

V'v = 0 in f~r, (2.5)

vlea = 0,

where rc is a pressure term associated with the irrotational vector field v - ~ Av - 0. From known existence and uniqueness results I-4, 2] we find that the field v exists and satisfies the estimate

I]vllk,.,+3,r < C [ l O I I k , ~ + l , r .

From Lemma 2.1 it is clear that v satisfies (2.1) and inequality (2.4). It is also clear, in view of the assumption on f~ and of the uniqueness of problem (2.5), that v is uniquely determined. []

Our next goal is to show solvability and obtain appropriate a priori estimates for certain initial-value problems. To this end we need a preliminary lemma on inequalities involving norms in Hm(f~).

Lemma 2.3. Let m > O. I f v (3. X m + 2 and u 6 Hm(~)), then a

I(v" Vu, u),,I < C1 IIv[l~+2[lu[l~. ( 2 . 6 )

I f v ~H"+3(f~) and u ~ Hm(~), then

I[u'Vvll,, < C2 [IvH,.+3 IlUlIm. (2.7)

I f v ~H'~+2 (f~) and u E/-/m+ 1 (f2), then

I[ v " V u llm <--_ C311V [lm+ z ll u llm+ l . (2.8)

Proof. Inequality (2.6) is proved as in [10]. Concerning (2.7), we notice that

Ilu'Vvll,, < C l l v l l c - + l L l u l l , .

and so, by the Sobolev embedding theorem [1], we deduce (2.7). Inequality (2.8) is proved in the same way. []

3Notice that if m = 0, the trilinear form (v. Vu, u) is identically zero.

302 G.P. GALDI 8r A. SEQUEIRA

We now consider the unique solvability of the initial-value problem

~u v #-~+-(u~ - e u r l v ) = u - V v - v . V u inl)T,

(2.9) U(', to ) = U(to) in ~.

Specifically, we have

Lemma 2.4. Assume that vsL~(I;Xm+3), m > 1, with Ilvll,.+3,T < M, and that U(to) ~ Hm(~). Then there exists a unique solution u to (2.9) such that

u ~ L~176 W ~' ~ ( I ; H m- ~(n))

d~t r n - l , T IIUI[m,T + < C, (2.10)

with C = C(f~,m,M, T,v,~, Ilu(to)14m). Moreover, if V'u(to) = 0, then V ' u = 0 in ~T.

Proof. Let us derive an a priori estimate for the solution of problem (2.9). To this end, we apply the derivative operator D k (k is a multi-index) to both sides of (2.9)1, take the scalar product in L2(O) with Dku and sum over k, with 0 < Ikl < m. We thus obtain

l d v 2dt [lul]Zm + V llullZ,, = -(eurlv, u)m + (u. Vv, u)m - (V. Vu, (2.11)

By using the Schwarz and Cauchy inequalities along with Lemma 2.3, we find that 2 I(eurlv, u),.I =<�89 + Ilull~),

I(u" Vv, U)m I < II U" Vv Itm It u I1~ --< C2 II v lira+ 3 II u It 2, (2.12)

I(v" Vu, u)ml <-_ Ca IlVllm+2 Ilull2m.

Thus, collecting (2.11), (2.12) we conclude that

l d ~ v 2dtl lul[2+ Ilult~ < ~llvll~m+l + (Ca + C2) llVllm+3llullZ,,. (2.13)

Integrating this inequality over I, with the help of Gronwall's lemma we find that

][UHm ~ D1, (2.14)

with D1 = D 1 (f~, m, M, T, v, a, II u (to)J] m). Moreover, from (2.9), with (2.14) and the estimates of Lemma 2.3, it also follows that

du <-V(IlulIm-~,T + IlVJlm. T) Z m - l , T = O~

+ C(llVllm+2, Tl[u]lm-l,T + I]vltm+I,TI[uL.,T) = D 2 (2.15)

with D2 = D2(~, m, M, T, v, ~, I] U(to)lira)-


With estimates (2.14) and (2.15) in hand, we can show by using the Galerkin method [18] that there exists a solution to problem (2.9), satisfying the regularity properties stated in the lemma, for all T > 0. The uniqueness of the solution follows easily from Gronwall's lemma.

Let us finally show that u is solenoidal in Dr, provided that U(to) has the same property. In fact, taking the divergence on both sides of (2.9)1 and using the general identity (1.3) we get

~ t + ~ = - V . ( v . V u - u. Vv) = - V . ( ~ v )

with { = V.u. Now, from the equality

v . ( ~ v ) = ( v . v ) ~ + ~ ( v - v)

it follows that

e--;+ = - ( v . v ) ~ , ~ ( . , t o ) = 0 ,

and we easily prove that the unique solution of this homogeneous transport equation is ~(t) = 0 for all t ~ I. The proof of the lemma is therefore com- pleted. []

The last result of this section problem (1.4)-(1.7) which for convenience we rewrite:

concerns a priori estimates for solutions to

in D•

~U V + ~ ( u - c u r l y ) = u ' V v - v ' V u O--T

c u r l ( v - ~ A v ) = u

V.v = 0

vla~ = 0, t ~ [-0, T],

u(x,O) = e u r l ( v o - ~ A v o ) - U o .

(2.16)

In this analysis we often need a simple result on a differential inequality, which we prove here in the form of a lemma.

Lemma 2.5. Let y( t ) be a smooth positive function in [0, T] satisfyin 9 the inequality

y'(t) + (kl - kzyP(t))y(t) < F( t ) for all t c I-0, T], (2.17)

where kl > O, kz ~ R, p > O and

T

F ( t ) a < oo. 0

304 G .P . GALDI 8r A. SEQUEIRA

Moreover, let ~ > 0 be such that k~ - k2a p = k > O. I f

T

0

then it follows that y(t) < e for all t ~ [0, T ] ,

Proof. Assume for contradic t ion that for some f we have

y ( f ) = e and y ( t ) < e V t e [ O , f ) .

Since k t - k2e p = k > O, we obta in the inequali ty

y'(t) + ky(t) < F( t ) Vt E [0, t ] ,

which when integrated over [0, f ] gives

f

y( f ) < y(O) + S F(s)ds; 0

hence g

y(f) ___ y(0) + (5 < 5 + (5 <

for (5 small enough. [ ]

L e m m a 2.6. Assume that (v, u) is a solution to (2.16) and that Uo e Vm, m > 1, with

u ~ L~176 (0, T ; H ' ( f ~ ) ) ~ W 1'* (0, T;Hm-l ( f~) ) ,

vEL~(O,T;X ,n+3) .

Then there exists (5 = (5 (f~, m, v, c~) > 0 such that if

II Uo lira _-< (5, (2.18)

then

T 2 [lUllm, T + S Ilu(s)ll~ds ~ (511juollL

0

where (51 and (52 depend only on ~, m, v, ~ and (5.

du < 62, (2.19) " -~ m - l , T

Proof. We replace u in (2.16)1 by curl(v - c~Av) and use the identi ty (1.3) to obta in

~tcurl(v -- c~Av) + V(eurl(v -- ~Av) -- curly) = eurl(curl(~ Av -- v ) x v). (2.20)

Setting ~ = curly and el iminating the curl on bo th sides of (2.20), we find that there exists a scalar field p such that

0 ~ ( v - c~Av) - vAv = (eAco - co) x v + Vp. (2.21)


Since V- v = 0, v [~n = 0, mult iplying (2.21) by v and integrat ing by par ts over f], we get

~ ( I g [ 2 -}-~x[Vl~[2)dx -]-- 2 v ~ ] V l p l 2 d x = 0 . ga

N o w we set

~o = min{1,~}, ~ = max{1 ,~}

and integrate over [0, t] to ob ta in

t C~ollvll~ + 2v~(IVvl 2 dx)(s)ds < ~l]lvo]l~

0 ~ Vt~EO, T].

F r o m the Poincar6 inequali ty (with cons tant 7) and with Vo = min { v, vy } we obta in

T C~o [[V[[2, T + VOS []r(s)][~ds < 2~1 11%1t2.

o

If we take fi = 2cq/min { v o, C~o }, then this implies that

T Itv[l~,r + ~ IIv(s)[]~ds </~llvo ]1~. (2.22)

0

N o w let us prove an energy est imate on u. Mult iplying (2.16)1 by u and integrat ing by par ts over f~, we easily find that

l d v v 2dt~n lu]zdx +-~ ~n eurlv'udx + nSu'Vv'udx

since it is clear that ~nv'Vu'udx = 0. Using the Cauchy inequality, the est imates of L e m m a 2.3 for m = 0, and the est imate (2.4) of L e m m a 2.2 (with q~ = u), we get

d v( d--t Ilull~ + 2vllutlgc~ <-- [Ivll~+ Ilullg) + 2Cllull 3

O~

and thus the differential inequali ty

d~ltullg+ -2CILu]lo Ilul/o z<-Ilvl[~,~ (2.23)

which we can write in the fo rm

y'(t) + ( V - 2cx~) )y ( t ) < F(t). (2.24)

r ight -hand side of (2.24) is control led by the initial da ta in the Since the sense that

T S F(s)ds < a (from the est imate (2.22)) 0

for some a > O, then according to L e m m a 2.5, there exist a l , a2 > 0 such that

Huo]lo < ~rl implies tha t Ilu(t)llo < a2 V t e [ 0 , T ] .


Using this last estimate together with (2.22) in inequality (2.23) yields

T 2 Ifullo. T + ~ Ilu(s)ll~ds < 0-3 Iluo II~ (2.25)

0

for some 0-3 > 0. Now, let us write (2.11) for m = 1 as

l d v v 2 dt II u II ~ + -~ It u II ~ = ~(eurlv, u)l + (u' Vv, u)t - (v. Vu,/g)l .

Using the estimates (2.12) for m = 1, we get

2dt llull d + v-Ilull~ < ~ (llvll~ + llull~) +

and consequently

(: ) v d~llull~ + -2C[Ivl l4 Ilull~ < -Ilvll~.e (2.26)

Now, by (2.25) and Lemma 2.2, we find that

T

S II v(t)II 22 at <= 0-3 II u0 II ~. 0

Therefore, by Lemma 2.5, (2.25), (2.26), and by a reasoning similar to that which we used before, it readily follows that there exist constants 04, 0-5 > 0 such that, if

Iluo 111 < 0-4, then

T

= k~ u(s)ll~ds < II u II 1, r + II = 0-5 II no II 0

with k > 0. Thus we conclude that the estimate (2.19)1 is valid for m = 1. We now prove the general case m > 2, by induction. Thus, assuming that (2.19)1 holds for m, let us show that it also holds for m + 1. By using the estimates (2.12) in (2.11) written for m + ! we deduce that

d-~llull~+l+ -2CI lu l lm+, Ilull2m+l_-<~ vl]~+2.

In view of Lemma 2.5 and of the inductive assumption, it follows, for [I Uo lira+ 1 sufficiently small, that

T

Ilutlim+,,r + j" Ilu(s)tI~+I ds < 0-6 Iluoll~+l. 0

Finally, we observe that the estimate (2.19)2 follows easily from (2.15), in view of (2.19)1. []

Remark. Concerning the two-dimensional case we must observe that the main results of the preceding lemmas remain essentially the same and are even stronger.


In fact, since the conditions determining the vector potential are more stringent than those defining the two-dimensional stream function, Lemmas 2.1 and 2.2 are still valid. On the other hand, in g 2 we rewrite problem (2.16) in the form

Ou V(u & + ~ - c u r l y ) = - v . Vu }

curl(v - c(Av) = u

V.v = 0

in f~ x (0, T),

vl~n = 0 , t e [0, T ]

u(x,O) = curl(v 0 - ~Avo) - Uo,

(2.27)

where u is now a scalar function. This system is obtained from problem (t.1), (1.2) by taking the scalar curl on both sides of the main equation and using the two-dimensional result

curl(curl(v - ~Av) x v) = v' Vcurl(v - c~Av).

Following the proofs of the preceding lemmas we observe that the results remain valid in the two-dimensional case, in spite of the absence of the term u ' Vv on the right-hand side of the system (2.27)1.

3. Global existence of classical solutions

In this section we prove the existence of a unique global solution for problem (1.1), (1.2) (equivalent to problem (1.4)-(1.7)) when the initial data are small enough in the following way. First, by the Schauder fixed-point theorem, we prove the existence of a local solution in I - (to, to + T) for all to >_- 0 where T depends on an upper bound for H U(to)lira, but is otherwise independent of to. Then using the global a priori estimates of Lemma 2.6 we can show the existence of a solution to (1.4)-(1.7) for all t > 0. We begin by recalling the following well- known result.

Theorem 3.1 (Schauder Fixed-Point Theorem). A compact mapping 4) of a closed bounded convex set G in a Banach space Y into itself has a f ixed point.

Take the Banach space

Y = C ( I ; V m - 1 ) , r e > l ,

and for D > 0 define

G = {(p �9 Y: q~ �9 L~176 [I (P Hm, T < D, (p(x, to) = u(x, to) �9 Vm}.

Consider now the map

308 G.P. GALDI 8,: A. SEQUEIRA

defined in G as the composition of the operator to ~ v defined by

carl(v -- ~Av) = to in ~ T ,

V ' v = O i n ' T ,

vl0a = 0

(3.1)

(where the time t is a parameter), with the operator v ~ u defined by

0u v & + ~ ( u - curly) = u . V v - v . V u in ~'~T,

(3.2) u(',to) = U(to) in ~.

Notice that proving the existence of a solution to our original problem (1.4)-(1.7) is equivalent to showing that the map

~b : G ~ Y - ~ Y

admits a fixed point. First we prove

Lemma 3.1. For all to and D the map �9 transforms the closed bounded convex set G into a relatively compact subset of Y. Moreover, for D sufficiently small, �9 is continuous in the topology of y.4

Proof. The closedness of set G is obvious. In fact, every sequence ton (n = 1 , . . . ) in G converging to to in Y has a subsequence to,, which converges weakly to a certain r in L~176 such that I[r T<liminfllto,k[lm, r. Thus Iltollm, T = Ij0 Nm, r__< D.

To prove the compactness property of W, let us denote by v. a sequence of solutions to problem (3.1), corresponding to the data to. e G, such that v. is uniformly bounded on I for the Hm+3(~)-norm (Lemma 2.2). Let u. e G be the associated sequence of solutions to problem (3.2). Since

un is bounded in L~(I; Vm),

dun dt

is bounded in L~(I; Vm-1),

we have in particular that un is bounded in WI'Z(1;Vm-1), and by classical compactness arguments we conclude that u, ~ u in Y.

To treat the continuity of �9 we still denote by u, u, ~ Y(n = 1 . . . . ) the corresponding images of to, to, s G(n = 1 , . . . ) under the map ~b. Let us subtract the

4The proof of continuity could be given without imposing restriction on D. However, this would be inessential for our purposes.


equations (1.9)1 written for u and u,, multiply the result by un - u and calculate the H m- l(f2)-inner product. Setting y(t) = 11 un - u lira- 1, after some easy calculations we get

y'(t) + 2y(t) < C(1 + ]lullm-l,T + IlUllm, r ) I I q , . - q, tl~,

where 2 can be chosen positive for D sufficiently small. Then noticing that ]l u H,, is bounded (Lemma 2.4) we conclude that [I u, - u Ilr < C1 II ~o. - ~o lit, and thus the continuity of 0Ii is proved. []

We are now ready to prove

Lemma 3.2 (Local Existence). Given arbitrary to > 0 and u(x, to) ~ Vm, m > 1, with ]1U(to) lira < D, D sufficiently small, there exists T > 0 such that problem (1.4)-(1.7) has a unique solution in [to,to + T ] with

(u, r) ~ G • (C(I; Vm+e)~L*(I;Hm+3(f~))) , (3.3)

dr ~ e L * ( I ; H m + 2 ( n ) ) .

In particular, if IlU(to)l[m < �89 then T can be chosen as

K1, / 1 + K2 T = -D-,n ~ ~9-+-~-2) > 0 (3.4)

where the positive constants K I and K2 depend only on f~, m, v and c~.

Proof. We apply the Schauder fixed-point theorem. In view of Lemma 3.1, for existence and the proof of (3.3)1 we only need to prove that ~b maps G into itself. Let us take ~o e G( [] 9 I1,., r _-< D) and fix to > 0. From (2.13), proceeding as in the proof of Lemma 2.4 and using Gronwall's lemma, we get

Nu[] am <= HU(to)llZ~e(G+C2)Mt+ vM

C2) (e(C1 + C2)Mt 1), ~(C1 +

where M = CD (with C the constant in Lemma 2.2). Thus, if ]1 u(x, to)Jim < D, it follows that there exists T > 0 such that ~b(G) c G and that, in particular, for ]1 u(x, to)lira < �89 we may take T as in (3.4). Condition (3.3)2 can be proved as in (2.15). Finally, the uniqueness of the solution follows from [5]. []

Using the global a priori estimates of Lemma 2.6, we can extend the local existence of the previous lemma to deduce our main result.

Theorem 3.2 (Global Existence). Let Uo~ Vm, m >_ 1. There exists eo = So (f2, m, v, ~) > 0 such that if

II no I1~ ~ ~o,


then problem (1.4)-(1.7) has a unique solution for all t e [0, oo ) with

v e C(O, T; Vm+2)c3L~176 T;Hm+3(f2))

dl~ ~ L~176 (0, T;H~+2(fl)),

for all T > O. Moreover, for m >-_ 2,

d2v dt ~ e L~(O, T ;H "+ l(f~)).

(3.5)

(3.6)

Thus in particular, for m = 4, v is a classical solution, i.e.,

v 6 C1(0, T; C3(f2)).

Proof. We choose lluo LI,, < min{D,6} = e. By Lemma 3.2 we have existence on [0, T]. Moreover, by Lemma 2.6 (inequality (2.19)1) we have s

I[u(Z)[/~ < Clluollm

for C independent of I[Uo lira and T. Choosing Iluo lira < t i c = to we get existence on IT, 2T] and again by Lemma 2.6 we get

Ilu(2Z)llm < Clluollm _-< to.

Repeating this procedure we obtain the solution on [0, + oo) which satisfies properties (3.5). It remains to show (3.6). Differentiating (1.5) with respect to time t, we get

02u v (c?u Or) a Vv) ~3 &--5- +-~ ~ - - eur l~ = ~ ( u ' - ~ (v .Vu) .

From Lemmas 2.6 and 2.2 we know that

u ~ L ~176 (0, T;Hm(~)),

Ou - ~ e L~(O, T ; H ~ - ~(fl)),

v ~ L~176 y;Hm+3(a)),

~v -~e L~176 T;tt'n+ 2(fl)),

SO that we have the estimates (cf Lemma 2.3)

< + o o , ~u Ilvlt.,+l,r+ Ilull,.-z,r N m+l,T _--< C1

v 'Vu) < C 2 Ilull,.-1,r + Ilvllm, r < + o o . m - 2 , T m , T m - l , T

sit is clear that [[u(T)mm is finite, since u e C(O, T; Vm_I)~L~176 T;Hm(f~)).


Hence we find that

~2 u

8t 2 - - E L~ (0, T ; H " - z ( ~ ) ) ,

and consequently from Lemma 2.2 we conclude (3.5). Now choosing m -- 4 we have

~2 u ~2V &--T ~ L ~ ( 0, T;H2(f~)) which implies that ~ - ~ L~(O, T;HS(f~)).

By a Sobolev embedding theorem we get

v~ W2 '~176 C~(O,T;C3(f~)). []

Remark. The results of Theorem 3.2 can also be obtained by using the Leray- Schauder fixed-point theory, as shown in a previous version of this paper [11]. However, the proof is more complicated than the one presented here.

Acknowledgments. We thank Dr. J. VlDEMAN for helpful discussions. This work has been supported by C.M.A.F./University of Lisbon, I.S.T. (Dep. of Mathematics)/Technical Uni- versity of Lisbon and MPI contracts 40% and 60%.

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Istituto di Ingegneria dell'Universit/~ di Ferrara Via Scandiana 21

44100 Ferrara

and

Departamento de Matemfitica Av. Rovisco Pais

1096 Lisboa Codex

(Accepted May 4, 1994)

further existence results for classical solutions of the equations of a second-grade fluid

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